Theorem bj-equsexvwd

Description: Variant of equsexvw . (Contributed by BJ, 7-Oct-2024)

	Ref	Expression
Hypotheses	bj-equsexvwd.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-equsexvwd.nf	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
	bj-equsexvwd.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	bj-equsexvwd	⊢ ( 𝜑 → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜓 ) ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-equsexvwd.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-equsexvwd.nf	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
3		bj-equsexvwd.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
4		alinexa	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ¬ 𝜓 ) ↔ ¬ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜓 ) )
5		bj-nnfnt	⊢ ( Ⅎ' 𝑥 𝜒 ↔ Ⅎ' 𝑥 ¬ 𝜒 )
6	2 5	sylib	⊢ ( 𝜑 → Ⅎ' 𝑥 ¬ 𝜒 )
7	3	notbid	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ¬ 𝜓 ↔ ¬ 𝜒 ) )
8	1 6 7	bj-equsalvwd	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → ¬ 𝜓 ) ↔ ¬ 𝜒 ) )
9	4 8	bitr3id	⊢ ( 𝜑 → ( ¬ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜓 ) ↔ ¬ 𝜒 ) )
10	9	con4bid	⊢ ( 𝜑 → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜓 ) ↔ 𝜒 ) )